1 On MultiCuts and Related Problems Michael Langberg Joint work with Adi Avidor On MultiCuts and Related Problems Michael Langberg California Institute of Technology Joint work with Adi Avidor

2 This talk Part I: Part I: Generalization of both Min. MultiCut and Min. Multiway Cut problems. Part II: Part II: Minimum Uncut problem.

3 Part I: Minimum MultiCut Input: Input: G=(V,E). G=(V,E).  : E  R +.  : E  R +. {(s i,t i )} i=1..k. {(s i,t i )} i=1..k. Objective: Objective: E’  E that disconnect E’  E that disconnect s i from t i for all i=1..k. s i from t i for all i=1..k. Measure: E’ of minimum weight. Measure: E’ of minimum weight. t3t3 s1s1 t1t1 s2s2 t2t2 s3s3 G=(V,E) MultiCut

4 Minimum Multiway Cut Input: Input: G=(V,E). G=(V,E).  : E  R +.  : E  R +. {s 1,s 2,…,s k }. {s 1,s 2,…,s k }. Objective: Objective: E’  E that disconnect E’  E that disconnect s i from s j. s i from s j. Measure: E’ of minimum weight. Measure: E’ of minimum weight. s4s4 s1s1 s6s6 s2s2 s5s5 s3s3 G=(V,E) Multiway Cut

5 Multicut vs. Multiway cut. Multicut: disconnect pairs {s i,t i } i=1.. k. Multicut: disconnect pairs {s i,t i } i=1.. k. Multiway Cut: disconnect {s 1,s 2,…,s k }. Multiway Cut: disconnect {s 1,s 2,…,s k }. NP-hard, extensively studied in the past. NP-hard, extensively studied in the past. Will present known results shortly. Will present known results shortly. Roughly: Roughly: Multiway Cut < Multicut. Multiway Cut < Multicut. Mutiway Cut: constant app. Mutiway Cut: constant app. Multicut: only logarithmic app. is known. Multicut: only logarithmic app. is known.

6 Our generalization: Minimum Multi-Multiway Cut Input: Input: G=(V,E). G=(V,E).  : E  R +.  : E  R +. {S 1,S 2,…,S k }: S i  V. {S 1,S 2,…,S k }: S i  V. Objective: Objective: E’  E that disconnect E’  E that disconnect all vertices in S i for i=1..k. all vertices in S i for i=1..k. Measure: E’ of minimum weight. Measure: E’ of minimum weight. s 11 s 23 s 12 s 13 G=(V,E) S1S1 s 21 s 22 s 24 S2S2 s 21 S3S3

7 Why generalization? s 11 s 23 s 12 s 13 G=(V,E) s 21 s 22 s 24 s 21 Input: Input: G=(V,E). G=(V,E).  : E  R +.  : E  R +. {S 1,S 2,…,S k }: S i  V. {S 1,S 2,…,S k }: S i  V. Multicut ({(s i,t i )} i=1..k ) Multicut ({(s i,t i )} i=1..k ) Each set S i ={s i,t i }. Each set S i ={s i,t i }. Multiway Cut: ({s 1,s 2,…,s k }) Multiway Cut: ({s 1,s 2,…,s k }) Singe set S 1 of size k. Singe set S 1 of size k.

8 Previous results Multicut MultiwayCut Multi-MultiwayCut {(s i,t i )} i=1..k {s 1,s 2,…,s k } {S 1,S 2,…,S k } APX-Hard [Dahlhaus et al.] APX-Hard [Dahlhaus et al.] APX-Hard [Dahlhaus et al] O(log(k)) [Garg et al.] O(log(k))  k [Cainescu et al. Karger et al, CunninghamTang] “Light inst.” log(Opt)loglog(Opt) [Seymore,Even et al.] --- “Light inst.” O(log(Opt)) + Our results

9 Results and proof techniques Multi-Multiway Cut results: Multi-Multiway Cut results: 4ln(k+1) approximation. 4ln(k+1) approximation. 4ln(2OPT) app. (edge weights  1).  4ln(2OPT) app. (edge weights  1).  Proof: Proof: Natural LP relaxation. Natural LP relaxation. Rounding: variation of region growing tech. [LeightonRao, Klein et al., Garg et al.] Rounding: variation of region growing tech. [LeightonRao, Klein et al., Garg et al.]

10 LP LP: Min:  e  (e)x(e) st: For every path P we want to disconnect  e  P x(e)  1  e  P x(e)  1 x(e)  0 Correctness: x(e)  {0,1} Correctness: x(e)  {0,1} s 11 s 23 s 12 s 13 G=(V,E) s 21 s 22 s 24 s 21 Multi-Multiway Cut P P

11 Rounding – region growing From LP: obtained fractional edge values. From LP: obtained fractional edge values. Implies a semi-metric on G. Implies a semi-metric on G. Simultaneously grow balls around vertices of connected sets until certain criteria. Simultaneously grow balls around vertices of connected sets until certain criteria. Each ball containes vertices close to center. Each ball containes vertices close to center. Remove all edges cut by balls. Remove all edges cut by balls. s 11 s 23 s 12 s 13 G=(V,E) s 21 s 22 s 24 s 21 Multi-Multiway Cut P1P1 Central: define the stopping criteria P2P2

12 Stopping criteria + analysis Based on that introduced by [GargVaziraniYannakakis]. Based on that introduced by [GargVaziraniYannakakis]. Consider both volume and cut value of union of balls. Consider both volume and cut value of union of balls. Main differences: Main differences: Simultaneously grow balls. Simultaneously grow balls. log(Opt): log(Opt): Change volume definition. Change volume definition. Grow large balls only. Grow large balls only. s 11 s 23 s 12 s 13 G=(V,E) s 21 s 22 s 24 s 21 Multi-Multiway Cut P1P1 P2P2

13 Part II: Minimum Uncut G=(V,E) Cut Input: Input: G=(V,E);  : E  R+. G=(V,E);  : E  R+. Objective: Objective: Cut Cut Measure: Measure: Minimum weight of uncut edges (dual to Min. Cut). Minimum weight of uncut edges (dual to Min. Cut). Find subset E’ of E of minimum weight s.t. G=(V,E-E’) bipartite. Find subset E’ of E of minimum weight s.t. G=(V,E-E’) bipartite.

14 Min. Uncut: previous results APX-Hard [PapadimitriouYannakakis]. APX-Hard [PapadimitriouYannakakis]. Min-Uncut < Min. MultiCut Min-Uncut < Min. MultiCut[KleinRaoAgrawalRavi]. App. ratio of O(log(|V|)). App. ratio of O(log(|V|)). Remainder of this talk: observations on attempt to improve app. ratio. Remainder of this talk: observations on attempt to improve app. ratio. G=(V,E)

15 Observations Our results imply: Our results imply: O(log(Opt)) approximation: If an undirected graph G can be made bipartite by the deletion of W edges, then a set of O(W log W) edges whose deletion makes the graph bipartite can be efficiently found. Min-Uncut < Min. MultiCut Min-Uncut < Min. MultiCut

16 Observations: LP Recall: Min. uncut has ratio O(log(n)). Recall: Min. uncut has ratio O(log(n)). Can show: Can show: Natural LP has IG  (log(n)). Natural LP has IG  (log(n)). LP enhanced with “triangle” constraints: IG  (log(n)). LP enhanced with “triangle” constraints: IG  (log(n)). LP enhanced with “odd cycle” con.: IG  (log(n)). LP enhanced with “odd cycle” con.: IG  (log(n)). LP combined with both: IG not resolved. LP combined with both: IG not resolved. LP: Min:  e  (e)x(e) st: For every odd cycle C,  e  C x(e)  1 st: For every odd cycle C,  e  C x(e)  1 triangle (metric):  i,j,k x(ij)+x(jk)-x(ik)  ≤ 1 triangle (metric):  i,j,k x(ij)+x(jk)-x(ik)  ≤ 1 odd cycle:  i 1,i 2,…,i l   j x(i j i j+1 )  ≥ 1 odd cycle:  i 1,i 2,…,i l   j x(i j i j+1 )  ≥ 1 x=1 x=0 x=1 1-x = metric

17 What about SDP? Natural SDP relaxation. Natural SDP relaxation. IG  (n). IG  (n). Adding triangle + odd cycle cons.: Adding triangle + odd cycle cons.: IG = ??? (relaxation is stronger than LP). IG = ??? (relaxation is stronger than LP). Standard random hyperplane rounding [GoemansWilliamson] : ratio =  (  n ½ ). Standard random hyperplane rounding [GoemansWilliamson] : ratio =  (  n ½ ). SDP: Min:  i  j  (ij)(1+x(ij))/2 st: X = [x(ij)] is PSD,  i x(ii)=1 st: X = [x(ij)] is PSD,  i x(ii)=1 triangle (metric):  i,j,k x(ij)+x(jk)-x(ik)  ≤ 1 triangle (metric):  i,j,k x(ij)+x(jk)-x(ik)  ≤ 1 odd cycle:  i 1,i 2,…,i l   j (1+x(i j i j+1 ))/2  ≥ 1 odd cycle:  i 1,i 2,…,i l   j (1+x(i j i j+1 ))/2  ≥ 1 x=1 x=-1 x=1

18 Concluding remarks Part I: Multi-Multiway Cut. Part I: Multi-Multiway Cut. Ratio that matched Min. Multicut O(log(k)). Ratio that matched Min. Multicut O(log(k)). Improve ratio for light instances O(log(Opt)). Improve ratio for light instances O(log(Opt)). Part II: Min. Uncut. Part II: Min. Uncut. Wide open. Wide open. Some naïve techniques don’t work. Some naïve techniques don’t work.